Very nearly all parasites are specific as to what they'll invade; so, no, you couldn't catch worms or anything like that.
I suppose it might be possible to pick up bacterial infections, like salmonella, but normal hygiene practices (washing your hands before and after working in your tank) would soon eliminate that risk.
As far as I am aware, the only really risky thin you can catch from your tank would be 'fish TB', or 'fish tank granuloma', which is really nasty and can take a long time to clear, but it's also vanishingly rare; even most doctors haven't heard of it. If you're really worried about it, don't put your hands in the tank if you have any broken skin (or get some of the long gloves that vets use) and don't mouth syphon; fill the tube by emersing it in the tank, and then take one end out with your finger or thumb over it.